400 XV NORMED ALGEBRAS AND SPECTRAL THEORY
For every eigenvalue a ofN,f(a) is an eigenvalue off(N), and the eigenspace
E(a; N) is contained in E(/(a);/(JV)).
Iff is continuous, then Sp(/(JV)) =/(Sp(J/V)).
(ii) More precisely, with the notation of (1 5.1 1 .3) , iffe *c(sP(^ii))»
f/ze spectrum off(Nn) consists of the complex numbers ft such that
essinf|/?-/(C)l =
C 6 Sp(JVn)
(relative to the measure un).
(iii) Iffe <%c(Sp(N)) fl/zrf z/# w a continuous mapping off(Sp(N)) into C,
(iv) 7f a sequence (fk) of functions belonging to ^c(Sp(JV)) ^ uniformly
bounded and converges simply tof, then for every x e E the sequence (fk(N) • x)
converges in E tof(N) • x.
We shall start by proving (ii). We have seen that/^J may be identified
with multiplication by the class of the function /in Lc(Sp(JVM), un). Hence
p $ SpC/X-AQ) if and only if there exists a real number a > 0 such that
N2(<j3-/>)^tf-N2(w) for all functions u e ^(Sp(^), /O (5.5.1). If
ess inf ]/? — /(Ol > ^> then we may take a to be equal to this number, by
C. e Sp(tfn)
virtue of (13.12.2), and therefore p $ Sp(f(Nn)). Conversely, if
essinf|/?~/(C)| = 0,
C e Sp(JVnX
then for each e > 0 the set M of complex numbers C e Sp(7Vn) such that
I0-/(OI ^'e is not ^-negligible, and we have N2(0? -f)(pM) ^ eN2(<pM),
hence /» e Sp(/(JVJ). _
It follows from (ii) that Sp(f(Nn)) c/(Sp(A^n)). If, moreover, / is con-
tinuous, then f(Sp(Nn)) is compact (3.17.9), and for each j5 =/(a), where
aeSp(Afn), every compact neighborhood of a has /vmeasure >0 (15.11.4),
so that/(Sp(J\g) = $p(f(NJ). To show that Sp(/(JV))c:/(Sp(N)) in general,
and that Sp(/(N)) =/(Sp(N)) when / is continuous, we have only to use
(15.11.5) and the fact that /(Sp(A/)) is compact if /is continuous. Finally,
the assertions about eigenvalues are evident, again by reducing to the case of
simple operators and using (15.11.6).
To prove (iii), note first that g °/is a universally measurable mapping of
Sp(JV) into C (13.9.6). The relation
follows from (15.11.1) in the case where #(Q = £p%q (/?, q being integers ^
Now, for each e>0, there exists a polynomial h in £ and £ such that
l<7(0 - h(Q\ £ e for all C e/(Sp(]V)) (7.3.2). Since |^(/(0) - A(/(0)| ^ e fore a, then